Higher-order pathwise theory of fluctuations in stochastic homogenization
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Higher-order pathwise theory of fluctuations in stochastic homogenization Mitia Duerinckx1,2 · Felix Otto3 Received: 7 March 2019 / Revised: 24 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order d2 , where d is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise proximity of the macroscopic fluctuations of a general solution to those of the (higher-order) correctors, via a (higher-order) two-scale expansion injected into the “homogenization commutator”, thus confirming the scope of this notion. This higher-order generalization sheds a clearer light on the algebraic structure of the higher-order versions of correctors, flux correctors, two-scale expansions, and homogenization commutators. It reveals that in the same way as this algebra provides a higher-order theory for microscopic spatial oscillations, it also provides a higher-order theory for macroscopic random fluctuations, although both phenomena are not directly related. We focus on the model framework of an underlying Gaussian ensemble, which allows for an efficient use of (secondorder) Malliavin calculus for stochastic estimates. On the technical side, we introduce annealed Calderón–Zygmund estimates for the elliptic operator with random coefficients, which conveniently upgrade the known quenched large-scale estimates. Keywords Stochastic homogenization · Higher-order homogenization · Linear elliptic equation · Fluctuations · Homogenization commutator
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Mitia Duerinckx [email protected] Felix Otto [email protected]
1
Laboratoire de Mathématique d’Orsay, UMR 8628, Université Paris-Sud, Orsay, France
2
Département de Mathématique, Université Libre de Bruxelles, Brussels, Belgium
3
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
123
Stoch PDE: Anal Comp
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 Higher-order theory of oscillations . . . . . . . . . 3 Higher-order homogenization commutators . . . . . 4 Reminder on Malliavin calculus . . . . . . . . . . . 5 Representation formulas . . . . . . . . . . . . . . . 6 Annealed Calderón–Zygmund theory . . . . . . . . 7 Proof of the higher-order pathwise result . . . . . . 8 Proof of the convergence of the covariance structure 9 Proof of the normal approximation result . . . . . . Appendix A. More Malliavin calculus . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
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